The Foundations of Mathematics

by Ian Stewart

The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory.

The authors have many years' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas.

This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups.

While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward.

This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.

Genres: Mathematics, Nonfiction, Science, Academic, Philosophy

414 pages, Kindle Edition

First published January 1, 1977

About the author

Ian Nicholas Stewart is an Emeritus Professor and Digital Media Fellow in the Mathematics Department at Warwick University, with special responsibility for public awareness of mathematics and science. He is best known for his popular science writing on mathematical themes.
--from the author's website

Librarian Note: There is more than one author in the GoodReads database with this name. See other authors with similar names.

Reviews

[Rida Hamadani · Rating 5 out of 5]

I was a high school freshman when I read this book, it is excellent!
I struggled as the pace of formalization of concepts accelerated, but with reading repetition, I grasped the wonderful ideas.
It is for those people who are very excited about pure mathematics, and can’t help but ask abstract questions all the time, such as “what is a number?”.
Probably the introduction to formal mathematics!

[Erickson · Rating 5 out of 5]

Very interesting and detailed in the proofs and explanations. However, for my standard, I need to read it again to gain fuller understanding. That's how I see this book.

[Westin Williams · Rating 2 out of 5]

Did Not Finish after 70 or so pages. This was not beginner-friendly or easy to follow as advertised. It takes things slow for the first 10 pages or so, then cranks out non-stop math and notation in a cramped format that is difficult to read. You definitely need to already be familiar with this kind of material beforehand. The math should have used much simpler examples and spaced things out more. As for the writing, it overexplained the obvious and underexplained the hard stuff. And the prose itself was just not very good or clear outside of the introduction sections.

Maybe I'm just dumb at math, but even if I were a math student, I think I'd find this book to be of limited use, as I'd probably would have learned these concepts more intuitively from a teacher.

[Indee3 · Rating 1 out of 5]

I really struggled to read this book. I understood maybe 20-30% of it. I often found that using additional things suh as youtube videos and wikipedia helped me to understand concepts but to me that defeats the purpose of having a single textbook to learn these ideas and i didnt always have intermet acess.

As someone who is a total begginer to university level maths I didnt think this book was of great benefit, im going to try some online moocs and other books whoch cover the same topics and when I do ill make sure to link/review them

[Lewis Waite · Rating 4 out of 5]

Heavy but was my sole source of prep for University level Maths

[Burke Peters · Rating 5 out of 5]

Valuable Brit views

[Fred Fidsol · Rating 3 out of 5]

ok

[Eleni · Rating 4 out of 5]

Very challenging.... But useful still. Should've really done the exercises at the end of each chapter though